(Normalized Euclidean distance):
Let \(E_b\)  be the average energy per information bit. The normalized Euclidean distance \(d^2(\mathbf {e})\)  of the error sequence \(\mathbf {e}\)  is defined as
\(d^2(\mathbf {e}) \triangleq \frac{1}{2E_b}\mathbf {e}^\text{H}\mathbf {G}\mathbf {e}\) .


(Operative Euclidean distance [1]}):
Denote by \(\mathbf {F}\)  the truncated version of \(\mathbf {G}\) , where \(\mathbf {F}\)  is a Toeplitz matrix constructed by circular shift of the vector \(\lbrace 0,...,0,g_{-L_E},...,g_0,...,g_{L_E},0,...,0\rbrace \) .
The operative Euclidean distance \(d_{\text{ope}}^2(\mathbf {e}) \)  of the error sequence \(\mathbf {e}\)  is defined as
\(d_{\text{ope}}^2(\mathbf {e}) \triangleq \frac{1}{2E_b}\mathbf {e}^{\rm {H}}\mathbf {F}\mathbf {e}\) .


(Hamming error weight):
The Hamming error weight \(w\)  is defined as the number of non-zero elements of an error sequence \(\mathbf {e}\) .


(Error probability for \(\mathbf {e}\)  [2]}):
Given the transmitted sequence \(\mathbf {x}\) , the probability of the ML detector declares the sequence \(\mathbf {x}^{\prime } = \mathbf {x} + \mathbf {e}\)  as the detection output is given by
\(P_{\mathbf {e}} = Q(\sqrt{\frac{d^2(\mathbf {e})E_b}{N_0}})\) .
